Chapter 1 Harmonic Oscillator Figure 1.1 illustrates the prototypical harmonic oscillator, the mass-spring system. A mass M is attached to one end of a spring. The other end of the spring is attached to something rigid such as a wall. The spring exerts a restoring force F = −kx on the mass when it is stretched by an amount x, i. e., it acts to return the mass to its initial position. This is called Hooke’s law and k is called the spring constant. 1.1 Energy Analysis The potential energy of the mass-spring system is U(x)= kx2/2 (1.1) k M F = − kx x Figure 1.1: Illustration of a mass-spring system. 1 CHAPTER 1. HARMONIC OSCILLATOR 2 U(x) turning points E K U x Figure 1.2: Potential, kinetic, and total energy of a harmonic oscillator plot- ted as a function of spring displacement x. which may be verified by noting that the Hooke’s law force is derived from this potential energy: F = −d(kx2/2)/dx = −kx. This is shown in figure 1.2. Since a potential energy exists, the total energy E = K +U is conserved, i. e., is constant in time. If the total energy is known, this provides a useful tool for determining how the kinetic energy varies with the position x of the mass M: K(x) = E − U(x). Since the kinetic energy is expressed (non- relativistically) in terms of the velocity u as K = Mu2/2, the velocity at any point on the graph in figure 1.2 is 1 2 2(E − U) / u = ± . (1.2) M ! Given all this, it is fairly evident how the mass moves. From Hooke’s law, the mass is always accelerating toward the equilibrium position, x = 0. However, at any point the velocity can be either to the left or the right. At the points where U(x) = E, the kinetic energy is zero. This occurs at the turning points 2E 1/2 xTP = ± . (1.3) k If the mass is moving to the left, it slows down as it approaches the left turning point. It stops when it reaches this point and begins to move to the right. It accelerates until it passes the equilibrium position and then begins to decelerate, stopping at the right turning point, accelerating toward CHAPTER 1. HARMONIC OSCILLATOR 3 the left, etc. The mass thus oscillates between the left and right turning points. (Note that equations (1.2) and (1.3) are only true for the harmonic oscillator.) How does the period of the oscillation depend on the total energy of the system? Notice that from equation (1.2) the maximum speed of the mass (i. 1/2 e., the speed at x = 0) is equal to umax = (2E/M) . The average speed must be some fraction of this maximum value. Let us guess here that it is half the maximum speed: 1/2 umax E uaverage ≈ = (approximate). (1.4) 2 2M However, the distance d the mass has to travel for one full oscillation is twice the distance between turning points, or d = 4(2E/k)1/2. Therefore, the period of oscillation must be approximately d 2E 1/2 2M 1/2 M 1/2 T = ≈ 4 =8 (approximate). (1.5) uaverage k E k 1.2 Analysis Using Newton’s Laws The acceleration of the mass at any time is given by Newton’s second law: d2x F kx a = = = − . (1.6) dt2 M M An equation of this type is known as a differential equation since it involves a derivative of the dependent variable x. Equations of this type are generally more difficult to solve than algebraic equations, as there are no universal techniques for solving all forms of such equations. In fact, it is fair to say that the solutions of most differential equations were originally obtained by guessing! We already have the basis on which to make an intelligent guess for the solution to equation (1.6) since we know that the mass oscillates back and forth with a period that is independent of the amplitude of the oscillation. A function which might fill the bill is the cosine function. Let us try substituting x = cos(ωt), where ω is a constant, into this equation. The second derivative of x with respect to t is −ω2 cos(ωt), so performing this substitution results in k −ω2 cos(ωt)= − cos(ωt). (1.7) M CHAPTER 1. HARMONIC OSCILLATOR 4 Notice that the cosine function cancels out, leaving us with −ω2 = −k/M. The guess thus works if we set 1 2 k / ω = . (1.8) M ! The constant ω is the angular oscillation frequency for the oscillator, from which we infer the period of oscillation to be T = 2π(M/k)1/2. This agrees with the earlier approximate result of equation (1.5), except that the approximation has a numerical factor of 8 rather than 2π ≈ 6. Thus, the earlier guess is only off by about 30%! It is easy to show that x = B cos(ωt) is also a solution of equation (1.6), where B is any constant and ω =(k/M)1/2. This confirms that the oscillation frequency and period are independent of amplitude. Furthermore, the sine function is equally valid as a solution: x = A sin(ωt), where A is another constant. In fact, the most general possible solution is just a combination of these two, i. e., x = A sin(ωt)+ B cos(ωt)= C cos(ωt − φ). (1.9) The values of A and B depend on the position and velocity of the mass at time t = 0. The right side of equation (1.9) shows an alternate way of writing the general harmonic oscillator solution that uses a cosine function with a phase factor φ. 1.3 Forced Oscillator If we wiggle the left end of the spring by the amount d = d0 cos(ωF t), as in figure 1.3, rather than rigidly fixing it as in figure 1.1, we have a forced harmonic oscillator. The constant d0 is the amplitude of the imposed wiggling motion. The forcing frequency ωF is not necessarily equal to the natural or resonant frequency ω = (k/M)1/2 of the mass-spring system. Very different behavior occurs depending on whether ωF is less than, equal to, or greater than ω. Given the above wiggling, the force of the spring on the mass becomes F = −k(x − d) = −k[x − d0 cos(ωF t)] since the length of the spring is the difference between the positions of the left and right ends. Proceeding as for CHAPTER 1. HARMONIC OSCILLATOR 5 k M d = d sin ω t 0 F x F = − k (x − d) Figure 1.3: Illustration of a forced mass-spring oscillator. The left end of the spring is wiggled back and forth with an angular frequency ωF and a maximum amplitude d0. the unforced mass-spring system, we arrive at the differential equation 2 d x kx kd0 + = cos(ω t). (1.10) dt2 M M F The solution to this equation turns out to be the sum of a forced part in which x is proportional to cos(ωF t) and a free part which is the same as the solution to the unforced equation (1.9). We are primarily interested in the forced part of the solution, so let us set x = x0 cos(ωF t) and substitute this into equation (1.10): 2 kx0 kd0 −ω x0 cos(ω t)+ cos(ω t)= cos(ω t). (1.11) F F M F M F Again the cosine factor cancels and we are left with an algebraic equation for x0, the amplitude of the oscillatory motion of the mass. Solving for the ratio of the oscillation amplitude of the mass to the am- plitude of the wiggling motion, x0/d0, we find x0 1 = − 2 2 , (1.12) d0 1 ωF /ω where we have recognized that k/M = ω2, the square of the frequency of the free oscillation. This function is plotted in figure 1.4. Notice that if ωF < ω, the motion of the mass is in phase with the wig- gling motion and the amplitude of the mass oscillation is greater than the amplitude of the wiggling. As the forcing frequency approaches the natu- ral frequency of the oscillator, the response of the mass grows in amplitude. CHAPTER 1. HARMONIC OSCILLATOR 6 2 1 x /d 0 0 0.5 1.5 2.0 ω /ω 0 F −1 −2 Figure 1.4: Plot of the ratio of response to forcing vs. the ratio of forced to free oscillator frequency for the mass-spring system. When the forcing is at the resonant frequency, the response is technically infinite, though practical limits on the amplitude of the oscillation will in- tervene in this case — for instance, the spring cannot stretch or shrink an infinite amount. In many cases friction will act to limit the response of the mass to forcing near the resonant frequency. When the forcing frequency is greater than the natural frequency, the mass actually moves in the opposite direction of the wiggling motion — i. e., the response is out of phase with the forcing. The amplitude of the response decreases as the forcing frequency increases above the resonant frequency. Forced and free harmonic oscillators form an important part of many physical systems. For instance, any elastic material body such as a bridge or an airplane wing has harmonic oscillatory modes.